1 00:00:07,164 --> 00:00:09,080 CHRISTINE BREINER: Welcome back to recitation. 2 00:00:09,080 --> 00:00:11,570 In this video, we want to work on using the change 3 00:00:11,570 --> 00:00:13,030 of variables technique. 4 00:00:13,030 --> 00:00:15,620 In particular, we're going to look at the following problem. 5 00:00:15,620 --> 00:00:18,880 It says, using the change of variables u is equal to x 6 00:00:18,880 --> 00:00:23,020 squared minus y squared and v is equal to y divided by x, 7 00:00:23,020 --> 00:00:26,970 supply the limits and integrand for the following integral, 8 00:00:26,970 --> 00:00:30,490 which is the double integral over region R of 1 over x 9 00:00:30,490 --> 00:00:32,380 squared, dx*dy. 10 00:00:32,380 --> 00:00:35,530 And R is the infinite region in the first quadrant that 11 00:00:35,530 --> 00:00:37,500 is both under the curve y equals 1 12 00:00:37,500 --> 00:00:40,000 over x, and to the right of the curve 13 00:00:40,000 --> 00:00:42,410 x squared minus y squared equals 1. 14 00:00:42,410 --> 00:00:44,360 So this is a challenging problem. 15 00:00:44,360 --> 00:00:46,030 Again, I want to point out we just 16 00:00:46,030 --> 00:00:47,780 want to find the limits and the integrand. 17 00:00:47,780 --> 00:00:50,820 You don't actually have to compute the integral. 18 00:00:50,820 --> 00:00:54,440 But it is going to be tough, but stick with it. 19 00:00:54,440 --> 00:00:57,870 Pause the video, give it your best shot-- hopefully 20 00:00:57,870 --> 00:01:00,900 you find the appropriate limits and integrand-- 21 00:01:00,900 --> 00:01:02,499 and then when you feel comfortable, 22 00:01:02,499 --> 00:01:04,790 bring the video back up, and I'll show you how I do it. 23 00:01:11,590 --> 00:01:12,760 OK. 24 00:01:12,760 --> 00:01:14,565 Welcome back. 25 00:01:14,565 --> 00:01:15,940 So once again, what we want to do 26 00:01:15,940 --> 00:01:17,900 is this change of variables problem 27 00:01:17,900 --> 00:01:20,750 where we've defined u to be x squared minus y squared, 28 00:01:20,750 --> 00:01:24,840 v to be y divided by x, and we have this region that 29 00:01:24,840 --> 00:01:26,560 is in the first quadrant and it's 30 00:01:26,560 --> 00:01:28,940 under the curve y equals 1 divided by x 31 00:01:28,940 --> 00:01:31,440 and it's to the right of the curve x squared minus y squared 32 00:01:31,440 --> 00:01:32,876 equals 1. 33 00:01:32,876 --> 00:01:34,250 And we want to compute the limits 34 00:01:34,250 --> 00:01:36,624 and integrand for that particular integral. 35 00:01:36,624 --> 00:01:38,290 So what I'm going to do, to try and make 36 00:01:38,290 --> 00:01:40,310 this as organized as possible, is 37 00:01:40,310 --> 00:01:44,300 I'm going to try first to graph the region R, 38 00:01:44,300 --> 00:01:47,214 or to figure out what the region R is in the xy-plane. 39 00:01:47,214 --> 00:01:49,380 Then I'm going to try and figure out what the region 40 00:01:49,380 --> 00:01:52,450 R is mapped to in the uv-plane. 41 00:01:52,450 --> 00:01:54,270 So what it looks like in the uv-plane. 42 00:01:54,270 --> 00:01:56,300 That will give me my limits. 43 00:01:56,300 --> 00:01:58,670 And then I'm going to try and determine the Jacobian. 44 00:01:58,670 --> 00:02:02,490 And then I will determine from that and the fact 45 00:02:02,490 --> 00:02:05,800 that I started with 1 divided by x squared as my function I was 46 00:02:05,800 --> 00:02:08,150 integrating, I will put those two together 47 00:02:08,150 --> 00:02:09,431 to figure out the integrand. 48 00:02:09,431 --> 00:02:11,430 So there are a bunch of steps to these problems. 49 00:02:11,430 --> 00:02:15,140 But the first one, again is I'm going to graph the region R. 50 00:02:15,140 --> 00:02:19,000 So I'm going to give you a very rough sketch, over here, 51 00:02:19,000 --> 00:02:21,569 of the region R. And I know it's in the first quadrant 52 00:02:21,569 --> 00:02:22,610 and I know it's infinite. 53 00:02:22,610 --> 00:02:25,380 I was already told that. 54 00:02:25,380 --> 00:02:25,880 OK. 55 00:02:25,880 --> 00:02:30,200 So in the xy-plane, the region R is below the curve y equals 1 56 00:02:30,200 --> 00:02:31,120 over x. 57 00:02:31,120 --> 00:02:32,510 So let me draw that curve. 58 00:02:35,710 --> 00:02:37,200 Again, this is very rough. 59 00:02:37,200 --> 00:02:38,370 This is a rough sketch. 60 00:02:38,370 --> 00:02:41,100 I'm putting up no scale on purpose. 61 00:02:41,100 --> 00:02:44,770 I'll put in one value, maybe, in this whole thing. 62 00:02:44,770 --> 00:02:45,610 OK? 63 00:02:45,610 --> 00:02:49,520 And so this is the curve y equals one over x. 64 00:02:49,520 --> 00:02:52,350 And then I need the curve which is 65 00:02:52,350 --> 00:02:55,400 part of the hyperbola that is x squared minus y squared 66 00:02:55,400 --> 00:02:56,280 equals 1. 67 00:02:56,280 --> 00:02:58,520 So I'll draw in the part we need, 68 00:02:58,520 --> 00:03:01,790 which looks roughly like this. 69 00:03:01,790 --> 00:03:03,490 Something like that. 70 00:03:03,490 --> 00:03:07,890 Again, this is not meant to be an exact graph, 71 00:03:07,890 --> 00:03:11,940 but to give you an idea of what the region looks like. 72 00:03:11,940 --> 00:03:14,790 And the only thing I'm going to mention 73 00:03:14,790 --> 00:03:18,500 is that this point we know is x equals 1 and y equals 0. 74 00:03:18,500 --> 00:03:20,520 So the region we're interested in that 75 00:03:20,520 --> 00:03:22,980 is both to the right of x squared minus y squared equals 76 00:03:22,980 --> 00:03:27,130 1 and below y equals 1 over x and in the first quadrant 77 00:03:27,130 --> 00:03:32,970 is exactly this region I'm shading here. 78 00:03:32,970 --> 00:03:36,350 So we want to understand what the values of u and v 79 00:03:36,350 --> 00:03:39,500 are along these bounds. 80 00:03:39,500 --> 00:03:42,650 We need to understand where this region maps 81 00:03:42,650 --> 00:03:44,820 to when I do the change of variables 82 00:03:44,820 --> 00:03:47,040 in order to understand what the limits are. 83 00:03:47,040 --> 00:03:51,770 So let me put the graph of this region in the uv-plane 84 00:03:51,770 --> 00:03:56,690 so that we can really understand what our bounds are. 85 00:03:56,690 --> 00:03:59,779 And I know already where it's going. 86 00:03:59,779 --> 00:04:02,070 So I'm going to just make the first quadrant, because I 87 00:04:02,070 --> 00:04:04,260 know this is going into the first quadrant. 88 00:04:04,260 --> 00:04:07,580 So it doesn't always work that something 89 00:04:07,580 --> 00:04:10,260 in the first quadrant maps into the first quadrant, 90 00:04:10,260 --> 00:04:13,860 but in this case, I already did the work, so I know it does. 91 00:04:13,860 --> 00:04:19,252 So let me point out a few things about where this region R maps. 92 00:04:19,252 --> 00:04:20,710 The first thing I want to point out 93 00:04:20,710 --> 00:04:23,800 is that we actually know that this curve, 94 00:04:23,800 --> 00:04:28,800 under the change of variables, maps to u equals 1. 95 00:04:28,800 --> 00:04:32,130 Because if you remember, u is equal to x squared minus y 96 00:04:32,130 --> 00:04:33,120 squared. 97 00:04:33,120 --> 00:04:37,740 So this whole curve is going to map to u equals 1. 98 00:04:37,740 --> 00:04:39,740 Now, I don't want the whole curve for my region. 99 00:04:39,740 --> 00:04:41,640 I only want this little piece of it. 100 00:04:41,640 --> 00:04:43,640 So I'm going to have-- in my uv plane, 101 00:04:43,640 --> 00:04:46,660 I'm going to have some segment at 1. 102 00:04:46,660 --> 00:04:48,810 And actually, I'll just know that it's 103 00:04:48,810 --> 00:04:52,980 some part of the line u equals 1 is going to show up in there. 104 00:04:52,980 --> 00:04:55,660 But if you notice, I know where it starts right away. 105 00:04:55,660 --> 00:05:00,200 Because at x equal 1, y equals 0, 106 00:05:00,200 --> 00:05:02,780 if I look at what v is-- if we come back here and remember 107 00:05:02,780 --> 00:05:06,790 what v is-- at x equal 1, y equals 0-- v is 0. 108 00:05:06,790 --> 00:05:10,990 And so my starting point on this segment-- if we come back here, 109 00:05:10,990 --> 00:05:12,760 my starting point on this segment 110 00:05:12,760 --> 00:05:16,760 is actually also at (1, 0). 111 00:05:16,760 --> 00:05:17,520 OK? 112 00:05:17,520 --> 00:05:19,550 So I know there's some point right 113 00:05:19,550 --> 00:05:25,520 here that maps down to here where the segment will stop. 114 00:05:25,520 --> 00:05:29,180 I'll find that point later, algebraically. 115 00:05:29,180 --> 00:05:29,680 Right? 116 00:05:29,680 --> 00:05:33,560 And then now we need to figure out where these two curves go. 117 00:05:33,560 --> 00:05:36,720 And then we can get a picture, and then 118 00:05:36,720 --> 00:05:38,220 we'll figure out what that point is, 119 00:05:38,220 --> 00:05:40,920 and we'll understand all the limits. 120 00:05:40,920 --> 00:05:44,800 So the first thing I want to point out is along this curve, 121 00:05:44,800 --> 00:05:49,469 we have y equals 0 and x is non-zero. 122 00:05:49,469 --> 00:05:51,010 And just to help ourselves, I'm going 123 00:05:51,010 --> 00:05:53,200 to rewrite what the change of variables 124 00:05:53,200 --> 00:05:57,720 is here, so I don't have to keep walking over to the other side. 125 00:05:57,720 --> 00:05:59,770 Our change of variables was u is equal to x 126 00:05:59,770 --> 00:06:03,420 squared minus y squared, and v was equal to y divided by x. 127 00:06:03,420 --> 00:06:08,130 So this whole curve has y equals 0. 128 00:06:08,130 --> 00:06:11,490 So what happens to u and what happens to v along that curve? 129 00:06:11,490 --> 00:06:14,470 Well, u is going to be x squared, 130 00:06:14,470 --> 00:06:16,760 and v is going to equal 0. 131 00:06:16,760 --> 00:06:18,710 And so the point of this, really, 132 00:06:18,710 --> 00:06:22,520 is that even though in u, this curve maybe 133 00:06:22,520 --> 00:06:25,890 is mapping at a different speed in some form to this curve 134 00:06:25,890 --> 00:06:30,030 here, it's still-- it's just taking that segment goes-- 135 00:06:30,030 --> 00:06:33,240 or that infinitely long ray goes to an infinitely long ray here 136 00:06:33,240 --> 00:06:34,900 along the u-axis. 137 00:06:34,900 --> 00:06:38,520 And again, that's because along this ray, y equals 0. 138 00:06:38,520 --> 00:06:40,920 And so v is equal to 0 everywhere on 139 00:06:40,920 --> 00:06:44,631 that ray and u is positive-- it's equal to x squared. 140 00:06:44,631 --> 00:06:45,130 OK? 141 00:06:45,130 --> 00:06:46,940 So I'm going to move the u out of the way, 142 00:06:46,940 --> 00:06:49,490 because we're going to say this is part of the region, 143 00:06:49,490 --> 00:06:51,880 or that's one bound of the region. 144 00:06:51,880 --> 00:06:54,490 And now I have to figure out where this curve goes. 145 00:06:54,490 --> 00:06:56,850 This curve is slightly more complicated, 146 00:06:56,850 --> 00:06:58,280 but I can still figure it out. 147 00:06:58,280 --> 00:07:02,110 So I'm going to show you how I do that sort of algebraically. 148 00:07:02,110 --> 00:07:04,590 That curve-- if you notice, if you remember-- 149 00:07:04,590 --> 00:07:07,580 is y equals 1 divided by x. 150 00:07:07,580 --> 00:07:10,550 So that means that on that curve-- 151 00:07:10,550 --> 00:07:14,740 let me even write it down-- on y equals 1 divided by x, v 152 00:07:14,740 --> 00:07:17,710 is equal to 1 divided by x divided by x. 153 00:07:17,710 --> 00:07:23,470 So v is equal to 1 divided by x squared, right? 154 00:07:23,470 --> 00:07:26,500 And then what does that mean about u? 155 00:07:26,500 --> 00:07:33,620 u, then, is equal to-- well, x squared is 1 divided by v, 156 00:07:33,620 --> 00:07:39,140 and then y squared, because y squared on that curve is just 1 157 00:07:39,140 --> 00:07:45,130 divided by x squared, is v. So let me just 158 00:07:45,130 --> 00:07:47,100 make sure we all followed that one more time. 159 00:07:47,100 --> 00:07:49,260 We're looking at where the curve y equals 1 160 00:07:49,260 --> 00:07:51,890 over x goes in the change of variables, right? 161 00:07:51,890 --> 00:07:53,950 So that's the top curve up here. 162 00:07:53,950 --> 00:07:57,572 y equals 1 over x is the top curve of our region R. 163 00:07:57,572 --> 00:07:59,030 So we want to know where that goes. 164 00:07:59,030 --> 00:08:01,120 Well, on y equals 1 over x, v is exactly 165 00:08:01,120 --> 00:08:03,690 equal to 1 over x squared, because v-- we know-- 166 00:08:03,690 --> 00:08:04,510 is y over x. 167 00:08:04,510 --> 00:08:07,870 So if I just substitute in for y, I get 1 over x squared. 168 00:08:07,870 --> 00:08:10,360 Now, if I look at this relationship, 169 00:08:10,360 --> 00:08:14,345 this means x squared is equal to 1 over v. So in terms of u, 170 00:08:14,345 --> 00:08:17,400 x squared becomes 1 over v. And then y 171 00:08:17,400 --> 00:08:20,120 squared-- which is 1 over x squared-- 172 00:08:20,120 --> 00:08:25,470 become v. So that curve is u equals 1 over v minus v. 173 00:08:25,470 --> 00:08:27,434 Now that curve, roughly, is going 174 00:08:27,434 --> 00:08:28,600 to look something like this. 175 00:08:35,330 --> 00:08:36,950 And it might seem strange. 176 00:08:36,950 --> 00:08:39,400 The thing is, I'm graphing this in the uv-plane, 177 00:08:39,400 --> 00:08:42,430 and I'm writing what looks like u as a function of v, 178 00:08:42,430 --> 00:08:44,870 and so it's sort of turned around from how you usually 179 00:08:44,870 --> 00:08:46,070 see these things written. 180 00:08:46,070 --> 00:08:51,480 But this is the equation that describes this curve. 181 00:08:51,480 --> 00:08:53,300 And that is sufficient to understand, 182 00:08:53,300 --> 00:08:57,500 because when we use our-- when we determine our bounds, 183 00:08:57,500 --> 00:09:01,190 we can determine our bounds from u equals 0 now, to u equals 1 184 00:09:01,190 --> 00:09:04,700 over v minus v. So we now have the bounds in u. 185 00:09:04,700 --> 00:09:06,630 We're actually doing quite well. 186 00:09:06,630 --> 00:09:08,830 So we have this region. 187 00:09:08,830 --> 00:09:11,430 We now have the bounds completely in u. 188 00:09:11,430 --> 00:09:16,100 u is going from u equals 0 to u equals 1 over v minus v. 189 00:09:16,100 --> 00:09:19,090 But the problem is now we don't know the bounds 190 00:09:19,090 --> 00:09:22,690 in v. We don't know what the bounds are in v, 191 00:09:22,690 --> 00:09:26,590 and so we have to be a little bit careful. 192 00:09:26,590 --> 00:09:27,387 So actually, no. 193 00:09:27,387 --> 00:09:28,220 I think I was wrong. 194 00:09:28,220 --> 00:09:29,230 It's not 0, is it? 195 00:09:29,230 --> 00:09:31,630 I said that twice now, and that was incorrect. 196 00:09:31,630 --> 00:09:36,220 u is going from 1, to 1 over v minus v. So I apologize. 197 00:09:36,220 --> 00:09:39,360 Because the slices of u are going from whatever 198 00:09:39,360 --> 00:09:41,480 the u-value starts-- which is at the value 1-- 199 00:09:41,480 --> 00:09:42,780 and it's coming this way. 200 00:09:42,780 --> 00:09:44,720 So I apologize. 201 00:09:44,720 --> 00:09:47,470 I was moving my arm like I was doing the v-values, 202 00:09:47,470 --> 00:09:50,040 but I actually wanted to do the u-values. 203 00:09:50,040 --> 00:09:52,340 So I want to go from where u starts-- which is at u 204 00:09:52,340 --> 00:09:57,030 equals 1-- to where u stops-- which is when it hits the curve 205 00:09:57,030 --> 00:09:59,440 1 over v minus v equals u. 206 00:09:59,440 --> 00:10:01,540 So hopefully I didn't confuse anyone by that. 207 00:10:01,540 --> 00:10:03,860 I'm glad I caught it, then. 208 00:10:03,860 --> 00:10:06,406 OK, so now we understand the bounds in u. 209 00:10:06,406 --> 00:10:07,780 And then to understand the bounds 210 00:10:07,780 --> 00:10:09,570 in v, all we need to understand is what 211 00:10:09,570 --> 00:10:12,850 is the v-value at this point. 212 00:10:12,850 --> 00:10:15,940 So once I know the v-value at this point, 213 00:10:15,940 --> 00:10:18,010 then I'm done with the bounds. 214 00:10:18,010 --> 00:10:20,220 So let's see if we can find that. 215 00:10:20,220 --> 00:10:24,500 Well, the v-value at that point is going to be at the point 216 00:10:24,500 --> 00:10:26,640 where these two curves intersect. 217 00:10:26,640 --> 00:10:32,019 So let's see if we can do a little algebra to understand 218 00:10:32,019 --> 00:10:33,060 what that will look like. 219 00:10:33,060 --> 00:10:35,660 So let me point out that where those curves intersect, 220 00:10:35,660 --> 00:10:42,250 I have the equation x squared minus 1 over x squared 221 00:10:42,250 --> 00:10:44,130 is equal to 1. 222 00:10:44,130 --> 00:10:46,980 And if I want to find x-values that satisfy this, 223 00:10:46,980 --> 00:10:49,836 I'm also looking for x-values that satisfy 224 00:10:49,836 --> 00:10:53,580 x to the fourth minus 1 is equal to x squared, 225 00:10:53,580 --> 00:10:57,900 which I can rewrite as x to the fourth minus x squared minus 1 226 00:10:57,900 --> 00:10:59,790 is equal to 0. 227 00:10:59,790 --> 00:11:04,010 So I can actually use the quadratic formula 228 00:11:04,010 --> 00:11:07,350 on this in terms of x squared. 229 00:11:07,350 --> 00:11:12,890 So what I get is I get x squared is equal to 1-- 230 00:11:12,890 --> 00:11:16,890 I get plus or minus root 5-- over 2. 231 00:11:16,890 --> 00:11:18,947 And if you look at it, the one you're actually 232 00:11:18,947 --> 00:11:21,280 interested in-- you can figure this out pretty quickly-- 233 00:11:21,280 --> 00:11:23,340 is the one that is plus. 234 00:11:23,340 --> 00:11:24,110 OK? 235 00:11:24,110 --> 00:11:27,080 I want the one that is plus root 5 over 2. 236 00:11:27,080 --> 00:11:33,090 So then that means x is the square root of this quantity 237 00:11:33,090 --> 00:11:35,740 at that point, right? 238 00:11:35,740 --> 00:11:37,600 Or I could actually think about it this way. 239 00:11:37,600 --> 00:11:39,300 Let me point out this. v is equal to 1 240 00:11:39,300 --> 00:11:42,330 over x squared at that point, because it's 241 00:11:42,330 --> 00:11:44,494 on that curve where we were talking about y 242 00:11:44,494 --> 00:11:45,160 equals 1 over x. 243 00:11:45,160 --> 00:11:46,940 So v is 1 over x squared. 244 00:11:46,940 --> 00:11:51,530 So 1 over x squared is just 1 over this quantity. 245 00:11:51,530 --> 00:11:52,950 So it's the reciprocal of this. 246 00:11:52,950 --> 00:11:57,640 It's also negative 1 plus root 5, over 2. 247 00:11:57,640 --> 00:11:59,460 You can check that if you need to. 248 00:11:59,460 --> 00:12:02,200 But I will write it down this way as the following: 249 00:12:02,200 --> 00:12:05,790 this is the point 1 comma a. 250 00:12:05,790 --> 00:12:12,530 And if I come over here, I will denote a will equal negative 1 251 00:12:12,530 --> 00:12:15,480 plus root 5, over 2. 252 00:12:15,480 --> 00:12:17,810 And that's really just 1 divided by x squared. 253 00:12:17,810 --> 00:12:20,620 So let me point that out again, that a 254 00:12:20,620 --> 00:12:24,987 is equal to 1 divided by x squared 255 00:12:24,987 --> 00:12:26,195 at the point of intersection. 256 00:12:32,220 --> 00:12:34,540 So hopefully you can see all that. 257 00:12:34,540 --> 00:12:38,410 So that tells us our bounds completely. 258 00:12:38,410 --> 00:12:40,480 We still have some work to do. 259 00:12:40,480 --> 00:12:42,020 So I'm going to put in the bounds 260 00:12:42,020 --> 00:12:44,259 and I'm going to leave an empty space. 261 00:12:44,259 --> 00:12:44,800 Actually, no. 262 00:12:44,800 --> 00:12:46,652 I won't do that, because this can get 263 00:12:46,652 --> 00:12:47,860 a little messy and confusing. 264 00:12:47,860 --> 00:12:49,402 So I'm just going to do the Jacobian, 265 00:12:49,402 --> 00:12:51,610 and then we'll figure it all out and write the answer 266 00:12:51,610 --> 00:12:53,760 right at the end, so there's no confusion. 267 00:12:53,760 --> 00:12:56,410 But hopefully you see at this point that we have the bounds. 268 00:12:56,410 --> 00:12:59,830 We know that u goes from 1, to 1 over v minus v. 269 00:12:59,830 --> 00:13:02,910 And v goes from 0 up to a, where a 270 00:13:02,910 --> 00:13:05,400 is the value I've written here. 271 00:13:05,400 --> 00:13:07,210 So we know the bounds. 272 00:13:07,210 --> 00:13:10,250 So now we have to figure out the integrand. 273 00:13:10,250 --> 00:13:14,350 So let's first compute the Jacobian, OK? 274 00:13:14,350 --> 00:13:19,940 So now we're looking at del u, v del x, y, 275 00:13:19,940 --> 00:13:23,410 using the notation we've seen in class. 276 00:13:23,410 --> 00:13:25,602 And so del u, v del x, y is going 277 00:13:25,602 --> 00:13:28,680 to be the determinant of the following matrix: 278 00:13:28,680 --> 00:13:30,690 2x, negative 2y. 279 00:13:30,690 --> 00:13:33,130 And then the derivative respect to v of x 280 00:13:33,130 --> 00:13:36,340 is negative y over x squared. 281 00:13:36,340 --> 00:13:39,810 And the derivative of v with respect to y is just 1 over x. 282 00:13:39,810 --> 00:13:49,850 So if I take the determinant of that, I get 2 minus 2 y 283 00:13:49,850 --> 00:13:52,600 squared over x squared. 284 00:13:52,600 --> 00:13:55,275 Which if you notice, in terms of our change of variables, 285 00:13:55,275 --> 00:14:04,370 is exactly equal to 2 minus 2 v squared, because v is y over x. 286 00:14:04,370 --> 00:14:06,340 And so I can rewrite this as 2 times 287 00:14:06,340 --> 00:14:10,010 the quantity 1 minus v squared. 288 00:14:10,010 --> 00:14:11,190 OK? 289 00:14:11,190 --> 00:14:13,910 So, so far so good, hopefully. 290 00:14:13,910 --> 00:14:18,900 Now let's figure out how to do the final step. 291 00:14:18,900 --> 00:14:21,780 So the final step-- I'm going to come back over 292 00:14:21,780 --> 00:14:24,795 and just remind us what the integrand was, OK? 293 00:14:24,795 --> 00:14:28,230 If we come over here, we're reminded that we were 294 00:14:28,230 --> 00:14:33,050 integrating over the region R of 1 over x squared, dx*dy. 295 00:14:33,050 --> 00:14:33,550 Right? 296 00:14:33,550 --> 00:14:35,760 That's what we were interested in initially. 297 00:14:35,760 --> 00:14:40,672 So now, if we come back, I'm going to write that down just 298 00:14:40,672 --> 00:14:41,755 to have it as a reference. 299 00:14:48,480 --> 00:14:50,130 OK, that's what we had initially. 300 00:14:50,130 --> 00:14:51,220 Let me make sure. 301 00:14:51,220 --> 00:14:52,950 Yes, that's what we had initially. 302 00:14:52,950 --> 00:15:03,030 And so now we know dx*dy is equal to du*dv over 2 times 1 303 00:15:03,030 --> 00:15:04,620 minus v squared. 304 00:15:04,620 --> 00:15:07,730 So that is going to replace the dx*dy. 305 00:15:07,730 --> 00:15:10,470 And now we have to figure out what to do with the 1 306 00:15:10,470 --> 00:15:12,530 over x squared. 307 00:15:12,530 --> 00:15:16,470 But, what do we have here? 308 00:15:16,470 --> 00:15:17,660 Now I have to remind myself. 309 00:15:17,660 --> 00:15:20,160 I can't remember all the steps anymore. 310 00:15:20,160 --> 00:15:26,710 We have u is equal to x squared minus y squared. 311 00:15:26,710 --> 00:15:27,470 Let me come back. 312 00:15:27,470 --> 00:15:28,970 Now I've forgotten what I was doing. 313 00:15:31,600 --> 00:15:32,260 Ah, yes. 314 00:15:32,260 --> 00:15:34,310 Now I remember, sorry. 315 00:15:34,310 --> 00:15:34,980 OK. 316 00:15:34,980 --> 00:15:37,960 So the point I should have remembered that I forgot, 317 00:15:37,960 --> 00:15:41,960 is that 1 minus v squared is equal to u 318 00:15:41,960 --> 00:15:42,927 divided by x squared. 319 00:15:42,927 --> 00:15:44,510 That's what I had figured out earlier, 320 00:15:44,510 --> 00:15:46,940 that I just forgot when I was staring at the board. 321 00:15:46,940 --> 00:15:49,530 And to notice that, what do we have to remember? 322 00:15:49,530 --> 00:15:52,730 u is x squared minus y squared, so if I divide everything 323 00:15:52,730 --> 00:15:54,220 by x squared, the first term is 1 324 00:15:54,220 --> 00:15:55,880 and the second term is v squared. 325 00:15:55,880 --> 00:15:57,879 So, whew, that's good. 326 00:15:57,879 --> 00:15:59,670 So I was a little nervous there for second, 327 00:15:59,670 --> 00:16:01,150 but I did in fact do this earlier. 328 00:16:01,150 --> 00:16:03,360 And I'd forgotten what I did. 329 00:16:03,360 --> 00:16:06,930 So now, the 1 minus v squared is actually the same 330 00:16:06,930 --> 00:16:09,970 as u divided by x squared. 331 00:16:09,970 --> 00:16:12,930 And notice what that does to this term here. 332 00:16:12,930 --> 00:16:20,770 That tells us that dx*dy over x squared is actually equal 333 00:16:20,770 --> 00:16:29,710 to du*dv over-- instead of the 1 minus v squared, 334 00:16:29,710 --> 00:16:32,610 I put u over x squared and I get-- notice, 335 00:16:32,610 --> 00:16:38,100 I get an x squared times 2, u divided by x squared. 336 00:16:38,100 --> 00:16:38,600 Right? 337 00:16:38,600 --> 00:16:42,070 I just replace the 1 minus v squared with what I know it is, 338 00:16:42,070 --> 00:16:48,604 the x squareds divide out, and so I get du*dv over 2u. 339 00:16:48,604 --> 00:16:50,520 So now the good news is I have all the pieces, 340 00:16:50,520 --> 00:16:52,470 because I'm about to run out of board space. 341 00:16:52,470 --> 00:16:53,690 So I have all the pieces, so I'm just 342 00:16:53,690 --> 00:16:55,800 going to put them together, and then we're done. 343 00:16:55,800 --> 00:16:58,620 So let me come here in the final spot, 344 00:16:58,620 --> 00:17:02,570 and say this is our final answer. 345 00:17:02,570 --> 00:17:08,070 Our final answer is that we're integrating u from 1, 346 00:17:08,070 --> 00:17:15,910 to 1 over v minus v. And then we're integrating v from 0 347 00:17:15,910 --> 00:17:19,350 to a-- where a is the value I determined earlier-- 348 00:17:19,350 --> 00:17:26,110 of 1 over 2u, du*dv. 349 00:17:26,110 --> 00:17:30,330 So this is the final, final answer. 350 00:17:30,330 --> 00:17:31,287 This was a long one. 351 00:17:31,287 --> 00:17:33,620 And I'm sorry I had a little brain freeze in the middle. 352 00:17:33,620 --> 00:17:36,060 I couldn't remember how I'd fixed that problem. 353 00:17:36,060 --> 00:17:38,140 So what I did at that point-- I just 354 00:17:38,140 --> 00:17:41,780 want to point out that when I was working on this problem, 355 00:17:41,780 --> 00:17:44,870 and I had a 1 minus v squared, I knew somehow 356 00:17:44,870 --> 00:17:47,180 I had to figure out how to relate that 357 00:17:47,180 --> 00:17:50,560 and the x squared in terms of u and v. 358 00:17:50,560 --> 00:17:53,010 And so I actually saw this expression. 359 00:17:53,010 --> 00:17:54,895 I could have written it better, maybe, as x 360 00:17:54,895 --> 00:17:56,970 squared times this equals u. 361 00:17:56,970 --> 00:17:57,470 OK. 362 00:17:57,470 --> 00:17:58,830 And maybe that would have been more obvious, 363 00:17:58,830 --> 00:18:00,020 if that's the case. 364 00:18:00,020 --> 00:18:02,130 But that was really the step that 365 00:18:02,130 --> 00:18:04,740 allowed me to replace all of this 366 00:18:04,740 --> 00:18:06,379 by things in terms of u and v. Which 367 00:18:06,379 --> 00:18:07,920 I know I should have been able to do, 368 00:18:07,920 --> 00:18:09,555 it's just a matter of figuring it out. 369 00:18:09,555 --> 00:18:11,180 So let me just go back to the beginning 370 00:18:11,180 --> 00:18:13,470 and remind you of each of the steps very briefly, 371 00:18:13,470 --> 00:18:15,900 and then we'll be done. 372 00:18:15,900 --> 00:18:18,510 So we come back over to the beginning. 373 00:18:18,510 --> 00:18:22,090 We were starting with change of variables supplied for us. 374 00:18:22,090 --> 00:18:24,510 We already had an integral in terms of x and y, 375 00:18:24,510 --> 00:18:26,560 and we had an infinite region. 376 00:18:26,560 --> 00:18:28,260 And what we were asked to do is find 377 00:18:28,260 --> 00:18:29,770 the limits and the integrand. 378 00:18:29,770 --> 00:18:33,170 So the first step for me is I always 379 00:18:33,170 --> 00:18:37,980 find it very helpful to draw the region in the xy-plane, 380 00:18:37,980 --> 00:18:40,500 and then draw the new region in the uv-plane. 381 00:18:40,500 --> 00:18:42,470 Neither one of them has to be perfect, 382 00:18:42,470 --> 00:18:48,750 but the understanding of the values of the curves in terms 383 00:18:48,750 --> 00:18:53,780 of equations of u and v are very important, to understand that. 384 00:18:53,780 --> 00:18:57,370 That gives you the bounds, the limits. 385 00:18:57,370 --> 00:18:59,797 And then, so we did all this work. 386 00:18:59,797 --> 00:19:00,630 We found the limits. 387 00:19:00,630 --> 00:19:02,338 There was a little algebra in the middle. 388 00:19:02,338 --> 00:19:03,270 We found the limits. 389 00:19:03,270 --> 00:19:05,780 And then we found the Jacobian, which 390 00:19:05,780 --> 00:19:09,640 was going to tell us how the variables were changing. 391 00:19:09,640 --> 00:19:11,480 We found it in terms of x and y. 392 00:19:11,480 --> 00:19:14,520 We rewrote it in terms of u and v. 393 00:19:14,520 --> 00:19:19,870 And so when we came back and we compared what our integrand was 394 00:19:19,870 --> 00:19:25,424 initially, we could compare dx*dy to du*dv. 395 00:19:25,424 --> 00:19:26,840 But then we also had to figure out 396 00:19:26,840 --> 00:19:29,170 how to replace the 1 over x squared. 397 00:19:29,170 --> 00:19:33,550 So once we did all that, we had everything in terms of u and v, 398 00:19:33,550 --> 00:19:36,045 and then we finally had what the integrand was going to be. 399 00:19:36,045 --> 00:19:38,265 So there were a lot of steps, but this was ultimately 400 00:19:38,265 --> 00:19:39,140 what the problem was. 401 00:19:39,140 --> 00:19:42,830 And again, I'll just point out, this is the final solution 402 00:19:42,830 --> 00:19:43,620 right here. 403 00:19:43,620 --> 00:19:47,780 We integrated from 1, to 1 over v minus v, for u. 404 00:19:47,780 --> 00:19:52,970 And we integrated from 0 to a in v, the function 1 over 2u. 405 00:19:52,970 --> 00:19:53,470 OK. 406 00:19:53,470 --> 00:19:55,497 That is where I will stop.